Numerical Simulation of Fluid Flow in a Dual Porosity Geothermal System with a Thin Zone of High Horizontal Permeability

\

NUMERICAL SIMULATION OF FLUID FLOW IN A DUAL POROSITY GEOTHERMAL SYSTEM WITH A THIN ZONE OF HIGH HORIZONTAL PERMEABILlTY~ /

By

Willis Jakanyango(Ambusso ~ Registration Number 184/0038/2003

A thesis submitted in partial fulfillment of the requirements for the award of the degree of Doctor of Philosophy in Physics of Kenyatta University.

September 2007

Ambusso, WiJlis

DECLARATIONS

This thesis is my original work and has not been presented for the award of a degree at anyother university.

Date

3.1.

..

.L.?.:~.7.

.

Willis J. Ambusso Signature :-:..

:'M~;='::::~.

This thesis has been submitted for examination with our approval as university supervisors.

Prof. I.V. S Rathore Supervisor

Physics Department Kenyatta University

Supervisor

Dr. A. S. Merenga Physics Department ,.. Kenyatta University

Signature ~

k; ~

Date

~.1{

..

~

.

Supervisor

Prof. J.P. Patel Physics Department University Of Nairobi

Signature ....•...

~.~.J

....

l.D.f.~t{);;;-DEDICATION

ACKNOWLEDGEMENTS

The duration of this study was one of immense personal enlightment and discovery during which I was privileged to work with and receive support from an array of committed scholars, administrators and staff at Kenyatta University. To all ofthem I am greatly thankful. However, I would like to single out a few of them. In particular I would like to express my gratitude to Prof. I.V.S Rathore for his guidance and unqualified support. To him Iam forever indebted and owe deep appreciation and respect especially for his encouragement, unreserved commitment, availability for all forms of consultations and assistance, and generosity with scientific and technical advice. I would also like to similarly thank Dr. A.S. Merenga for "starting it all" and most of all for identifying my interests and abilities from the very beginning. His knowledge of dual porosity systems proved

invaluable during the crucial parts of this study. I would also like to thank Prof. J.P. Patel of the Physics department of the University of Nairobi, for his advice on geothermal modeling and regional geology of the East African Rift Valley. I am equally grateful to Messrs. J. Gachuhi, S. Njuguna and F. Mudimba for providing the electronic and associated materials needed for this study.

,..

Financial support for this research was provided by Kenyatta University under the staff development program and the school board of The School of Pure and Applied Sciences of Kenyatta University. This generous support is gratefully acknowledged. The field data used in this study was provided by the Kenya Electricity and Generation Company

(KENGEN). This too is acknowledged.

ABSTRACT

A numerical simulator capable of mode ling fluid flow in layered dual porosity

geothermal systems with high horizontal permeability has been developed. The simulator can match multiple-peaked tracer profiles from injection tests and reliably predict

temperature and pressure changes caused by injection of brine in these systems. The simulator was tested and validated using data from Svartsengi geothermal field in

Iceland, a non-layered geothermal field, and Olkaria geothermal field in Kenya, a layered geothermal reservoir. Simulated results show significant improvement over regular geothermal simulators and indicate that fluid flow within the horizontal layers with high permeability are best represented by fracture flow theory where fluid flow between the matrix-fracture network occurs in both directions rather than what is normally assumed that fluid flow is governed purely by dual porosity model where fluid flow occurs only from the matrix to the fracture. The study shows that fluid movement in horizontal fractures will dominate main fluid flow features in the reservoir and shall uniquely modify tracer profiles. These horizontal fractures will also lead to high returns of injected fluid at relatively low speeds in reservoirs with moderate permeability as has been noted in Olkaria. This simulator makes it possible to extract quantitative values of the

TABLE OF CONTENTS

DECLARATION DEDICATION

ACKNOWLEDGEMENTS ABSTRACT

TABLE OF CONTENTS KEYWORDS

MATHEMATICAL SYMBOLS

11 111 iv

v

1.1 Background of Research 1.2 Objective ofthe study 1.3 Justification of study

CHAPTER 2 LITERATURE REVIEW

4.1 Introduction

4.2 Program Structure 4.3 Input Data

4.4 Procedure Code 4.4.1 Functions 1 4 9 10 11 11 13 14 23 23 25

28

48

48

55

2.2 Previous Studies Of Olkaria Geology 2.3 Numerical Simulation Of Olkaria CHAPTER 3 THEORY

3.1 Modeling of Geothermal Systems 3.2 Over view of the problem

3.3 Fundamental Equations 3.4 Discretization of equations

" 3.5 Method of Solution 3.5.1 Solution Development

3.5.2 Gauss-Seidal Iterative Method

CHAPTER 4 MATERIALS AND METHODS

Figure 3.3. Actual block surface arrangement in Cartesian System

39

CHAPTER 4 MATERIALS AND METHODS

Figure 4.1. Master flow chart

Figure 4.2. Main Data arrays

Figure 4.3 Summary of computation routines

50

57

60

CHAPTER 5 RESULTS

Figure 5.1 Vertical Pressure Profile

65

Figure 5.2 Schematic of an aquifer between two impermeable layers

66

Figure 5".3 Location of injection and production wells in the middle layer

67

Figure 5.4 Early Pressure changes in doublet system

68

Figure 5.5 Pressure changes for unequal doublet system

69

Figure 5.6 Effect oflnjection rate on temperature

71

Figure 5.7 Effect oflnjection rate on Pressure

71

Figure 5.8 Temperature changes between injection and production wells

72

Figure 5.9 Temperature changes around the injection well

73

Figure 5.10 Temperature changes between injection and production wells

74

Figure 5.11 Schematic of injection and production wells in the base case

78

Figure 5.12. Base tracer profile

79

Figure 5.13. Late time tracer profile for the base case

82

Figure 5.14. Effect of boundary on tracer profiles

83

Figure 5.15. Effect of porosity on tracer profiles

85

Figure 5.16 Effect of diffusion coefficient on tracer profile

87

Figure 5.17. Effect of injection rate on tracer concentration

89

Figure 5.18.

Combined tracer profile

95

Figure 5.19. Tracer profile for the matrix

96

Figure 5.20. Tracer profile for the thin layer

96

Figure 5.21.

Tracer profile for distant blocks

97

Figure 5.22 Tracer profiles for blocks nearest the injection well

100

Figure 5.23. Tracer profiles for blocks at intermediate distance

101

Figure 5.24.

Tracer profile for distant blocks from injection well

101

Figure 5.26 Location of wells in Svartsengi geothermal field

Figure 5.27 Tracer profile for Well 6

Figure 5.28 Matching normalized tracer profiles for well 6

Figure 5.29 Layout of Olkaria wells

Figure 5.30 OW-706 Tracer Profile

Figure 5.31

OW-32 Tracer profile

Figure 5.32 OW-706. Matching tracer profile

Figure 5.33 Matching tracer profile for OW-32

105

107

108

110

111

113

114

115

LIST

OF TABLES

CHAPTER 4 MATERIALS AND METHODS

Table 4.1. Summary of main functions

CHAPTER 5 RESULTS

Table 5.1

Simulated vertical pressure profile

65

Table 5.2 Temperatures at different injection rates

70

Table 5.3 Properties of common tracers and their usage'

76

Table 5.4 Tracer concentrations

81

Table 5.5 Summary of main parameters on effect of porosity

85

Table 5.6 Summary of main results on the effect of diffusion coefficient

87

Table 5.7 Summary

of main data on effect of injection rate

89

Table 5.8 Parameters for the base case studies

94

I'

Table 5.9 Summary of main results for multi-layered tracer profile

97

Table 5.10 Tracer arrival and Peak times for different layers

98

Table 5.11 Effect of Diffusion coefficient on tracer arrival and peak times

100

Table 5.12 Well data for Svartsengi Geothermal field

106

KEYWORDS

Drawdown - reduction or decline in stored mass or temperature or pressure or any other variable associated with the reservoir. Examples thermal, pressure, mass etc.

Formation - Extensive subsurface material consisting of rocks and fluids stored in the rocks.

Fracture - Substantial continuous cracks or breaks in a rock.

Geothermal system - Deep accumulation of fluids in formations at high temperature. Matrix - Single continuous body of porous rock.

Permeability - A measure of the ability of rocks to transmit fluid. Porosity - The fraction of void volume within a rock.

Saturation - Volume fraction of the void space occupied by a given phase of fluid.

Reservoir - substantial volume of subsurface fluid accumulation in porous rock. Removal or addition of large quantities of fluid will not affect total volumes in the reservoir.

Tracer- Chemically distinct substance that can be detected at low concentrations.

changes in

Mathematical symbols

cD

cD)

!! Viscosity

c Compressibility, Fractional change in volume per unit increase in pressure (inverse pressure, l/bar)

C Heat capacity per unit mass, (J/g or KJ/Kg) p Density, Mass per unit volume (Kg/M3)

<pC> volumetric heat capacity ofthe rock and fluid it contains, (J/cm3 or JIM3 ) u Internal energy of fluid phases per unit mass (j/cmIor KJ/Kg)

Urn Internal energy per unit mass of rock.

CHAPTER 1 INTRODUCTION

Thisstudy developed anumerical simulator for modeling fluid flow in layered dual

porosity geothermal systems in which significant permeability occur along horizontal

contactzones between geologically different formations. Contact zones between distinct

geologic formations are known to play a significant complimentary role tothe main

formations influid delivery in layered geothermal systems. However, these zones have

not been incorporated in geothermal simulators and are treated collectively with the bulk

permeability. This study sought to investigate techniques by which these important fluid

channels could be incorporated into numerical simulators. The simulator developed in

this study is capable of giving reliable interpretations of injection and tracer tests for

these reservoirs through deduction of key hydraulic parameters. This constitutes an

additional refinement in the interpretation of tracer tests and should enable better prediction ofresponse of layered geothermal systems to long-term steam production

accompanied by brine re-injection.

Conventional simulators incorporate fracture permeability through the use of double porosity. These simulators do not adequately give re-producible predictions for reservoirs

with additional permeability along the contact point between layers. This development hastherefore overcome a significant limitation of the presumed physical models that are

currently in use. Achieving areliable and usable simulation capability that can be applied

The development and calibration of this simulator relied heavily on data from Olkaria

geothermal field to test and validate results and enhance the simulators use for layered

reservoirs. A limited amount of published data on Svartsengi geothermal field in Iceland

wasused to test the ability of the simulator to model fluid flow in conventional non layered reservoirs (Gudmundsson and Hauksson, 1985). Olkaria geothermal field which islocated in the Kenyan rift valley was the first geothermal field to be developed for commercial power production in Africa and has been generating electricity for over twenty-five years (Mwangi and Simiyu, 2003). During this period the reservoir has experienced significant changes in pressure and temperature accompanied with phase

change of the dominant fluid types in the reservoir. The current thermodynamic parameters are different from those that prevailed at the start of production with the dominant phase in the reservoir changing from water to steam in several parts of the field (Mburu 2003, Kariuki 2004, Ofwona 2004a). Production wells have also under gone major changes in characteristics with a number of wells that previously produced two phase fluid now producing only dry steam while others have seen increase in water flow that is often accompanied with decline in steam flow (Ouma 2002, Ofwon a 2004b). These changes are due to fluid depletion and de-pressurization of the reservoir which still

,-has large heat reserves.

In the past decade the operating utility has undertaken studies aimed at establishing ways of reversing some of the effects of past production so as to extend the power production

lifeof thefield bytwo or more decades (Mwangi and Simiyu, 2003). Field tests, mainly injection and tracer tests have provided vital clues about the subsurface reservoir

structure by confirming some previously known properties and revealing new ones (Ouma,2002). All these have led to the development of large and valuable database for the field. This data has shown that there is a significant variation in some of the earlier

assumptions on the reservoir structure and raised questions on the whether regular reservoir simulators are applicable to the field.

Over the last two decades other countries transacted bythe African rift valley have also engaged inefforts to explore and develop geothermal energy. Wells drilled in these fields,which are all in the East African rift valley, have revealed that these geothermal

fields share the same fundamental geological structure with Olkaria (Jalludin, 2003,

Msonda, 2003, Teklemariam, 2004). In these fields different formations are layered on top of each other as seen in Olkaria geothermal field and are therefore likely to

experience the same changes as in Olkaria during production. Thus the simulator

developed here is likely to be of broader use and application in other fields in the region.

Apart from mode ling layered geothermal systems it was equally important to show that

this simulator could be applied to conventional non-layered reservoirs. Reservoirs of this

type ate more common and occur in many parts of the world. Svartsengi geothermal field

in Iceland was selected for this purpose. Limited amount of injection and tracer data on thisfield was found in published literature and was used to validate aspects of the simulator that apply to geothermal fields in which geological formations are not layered asis the case in Olkaria. Svartsengi geothermal field is not only none-layered but, does

not seem to be confined by a caprock as Olkaria is, and high temperatures are measured

close to ground surface (Gudmundsson and Hauksson, 1985). Several injection and tracer

testshave also been undertaken and reported in literature and provided vital data for

comparison with Olkaria.

1.1

BACKGROUND OF RESEARCH

This research was motivated by two main issues;

1. The immediate experience of injection and tracer tests in Olkaria. These tests

were done to investigate methods of restoring the field's steam production and

extending the life of the field, and,

2. The similarity in the geological structure of Olkaria geothermal field and other

fields in the Rift Valley.

Injection experience in Olkaria shows that the use of regular double porosity models in

the interpretation of injection and tracer tests does not give results that are consistent with

predictions from these models (Karingithi, 1993). Data obtained from such tests do not fit

theassumed model of the field and have always under predicted fluid return volumes

while over predicting benefits from these tests (Bodvarsson and Pruess, 1987, Ouma

2002). 'This has brought into question the underlying structure assumed in the models and

demanded a change and review of the presumed models that can be applied to these

fields.

Increased exploration in the greater Olkaria field and drilling results from new fields in

othercountries along the East African rift valley has led to an increase in the

understanding of the geological and thermal structure of the field. This information

which will probably experience similar production and injection trends as those seen in Olkaria. The development of this simulator will therefore find application in these fields as well.

Sinceinception of the Olkaria power plant in 1981 numerical simulations have been done periodically as part of the resource management program of the geothermal field

(Bodvarsson and Pruess, 1981, 1987; Bodvarsson, 1994). These simulations are done so asto augment field operation and development decisions such as power plant

enlargements and implementation of large-scale injection programs (Svanbojomson et al. 1983,Bodvarsson et al, 1986, Ofwona, 2003). Reservoir numerical simulations have not

onlyprovided vital clues on management possibilities but are the only way the many input scenarios can collectively be integrated to weigh each of the equally many output scenanos.

Thefirst large scale numerical simulation for Olkaria geothermal field after it

commenced production was done in 1987 (Bodvarsson and Pruess, 1987). This was two

years after the installation of the third turbine, which increased power generation from 30 Mwe

to

installation of the third turbine. A few months after the installation of the turbine several

centrally located wells experienced pressure transients and were unable to sustain the 6 barswell head pressure required at the power plant. This led to the lowering of the turbine inlet pressure from 6 bars absolute to 4 bars at which the power plant is still

operating to-date. The pressure transients resulted in a number of wells in the central part

production accompanied with increase in discharge enthalpy. The numerical simulation

byBodvarsson and Pruess (1987) was intended to investigate the cause of the transients

and asses whether they were of permanent or temporary nature. In addition the simulation

wasalso intended to seek ways by which the transients could be remedied or reversed.

One proposal put forward and tested by the study was re-injection of waste brine into

wells that were not connected to the power plant. This, it was presumed, would replace

thelostfluid andsupport reservoir pressure (Bodvarsson and Pruess, 1987). The

simulation was performed using the TOUGH-2 simulator developed by the Earth Science

division at the Lawrence Berkeley National Laboratory in California, United States of

America. TOUGH-2 isa multipurpose general simulator capable of simulating transport

ofmass and heat in heterogeneous porous media. It uses conventional reservoir structure

tosolve the fundamental equations and has no special features to account for extensive

lateralfractures such as those encountered in Olkaria.

Results of this simulation did show that brine re-injection would greatly benefit the

reservoir by offering both pressure support and replacing lost mass. The later would lead

directly to better heat extraction, extend life of the reservoir by several years and even

reduce the number of make up wells that would be needed to maintain the amount of

steam required to run the plant at the full capacity of 45 Mwe. Considering the use of

cold water as source of injection into one of the injection wells in the field, the simulation

predicted that a number of wells close to the injection well would experience immediate

increase in steam flow accompanied with enthalpy reductions. The simulator also showed

experience the same benefits, but, would overall still result in substantial benefits in most

parts ofthe field.

Injection tests were later designed and implemented to verify these predictions or

otherwise and serve as a prelude to full time injection as recommended bythe study

(Karingithi, 1993, 1995). To further improve the value of the tests tracers were added to

theinjected water to indicate the return paths of injected fluid.

Theearly fluid return speed for injected fluid was 2 meters/hour which were slow as

expected for a low permeability field like Olkaria (Karingithi 1993). However, the overall

fluidand tracer return volume of 31% was large and indicated that the reservoir had better connections between the wells than had been postulated. During the tests some

wells experienced changes in production as a result of the injection. However, this was

only after several months of injection, a much longer period than predicted by the

simulation. The tests did indicate that even cold water injection would benefit the field.

However this was due to reduction in the annual decline rates of steam production by

sustaining steam production rather than direct increase in steam flow rate that was predicted by the numerical simulation. A small number of wells close to the injection

,.

well experienced minor enthalpy decline but this was due to marginal increases in water

flow rates which were already very low at the time of the tests because of the natural decline that had taken place over the years. The magnitudes ofthese enthalpy changes

were smaller than those predicted bythe simulation. For many wells the steam flow rates

remained constant and at times even declined over short periods because of the increase

in watersurges (Karingithi, 1993).

Theseresults revealed that there were fundamental differences between the actual

reservoir structure and the one that was used in the numerical simulation: These

differences were due to differences in the mathematical implementation of the physical

model rather than pure error in either of them. The delayed response of the production

wellsto injection, the smaller than predicted changes in enthalpy and the contradictory

resultsfrom the tracer tests all indicated that the actual structure of the reservoir had not

been captured by.the simulation and required additional specifications. Field tests have

shown that conventional simulators that employ only dual porosity and oblique fractures

oflimited extent cannot accurately predict individual well production and overall

reservoir behavior characteristics. In case of Olkaria these simulators have over predicted

thenet enthalpy change and pressure recoveries while at the same time under predicted

therates of return of injected fluids and the overall fluid recoveries.

Other follow up simulations have been done since then but none of them have addressed

the inconsistencies between numerical predictions of injection tests and actual field tests.

This study aimed to address this issue by incorporating the key geological distinction

Olkaria field has over other fields in the world, namely, extensive horizontal fractures.

Whereas the role of horizontal fractures in the productivity of the wells in Olkaria was

recognized early during the exploration phase (Svanbojomson et al 1983; Odongo, 1986),

the full impact of these fractures was not fully appreciated until injection tests were done

in Olkaria in the early nineties. Resolving these seemingly opposing conclusions formed

a major motivation for this study.

The second reason for this study comes from drilling results in other geothermal fields in

Thesecountries are at various stages of geothermal development ranging from

geophysical exploration, to well testing and pilot power production. Wells in these fields

reveal a layered geologic system with a number of similarities infundamental geology and well characteristics (Teklemariam, 2004, Jalludin, 1003, Msonda, 2003). These

layered geothermal systems were probably formed from similar geologic episodes as

Olkaria.The contact points between these formations are the main sources of

permeability in these fields and deliver significant amount of fluid to wells, as is the case

inOlkaria (Teklemariam, 2004). These contact points act as horizontal fractures between

matrix formations with low permeability and are responsible for the keyproperties of the

these fields.

From the foregoing it is evident that a new approach to the problem of geothermal

numerical simulation has to be adopted in a way that will capture the fundamental modes

of fluid flow in layered reservoirs. These models should be better able to capture and reproduce these processes so as to predict future behavior more accurately than the

conventional simulators currently in use.

1.2 0BJECTIVE OF THE STUDY

The objective of this study was to develop a numerical simulator capable of mode ling

fluid flow in geothermal systems in which significant horizontal permeability occurs

along the contact zone between different geological formations.

The following were the specific objectives of this study:

1. Derive equations governing simultaneous flow of steam and water, energy

transfer and tracers in a system with the layered permeability.

2. Prepare and implement a computer program to numerically solve the equations in

1. that will incorporate injection in these reservoirs.

3. Perform matching of injection and tracer tests using the simulator to validate the

simulatorand determine optimum production and injection rates.

4. Identify other areas of geothermal mode ling where the simulator can be applied

and possible further developments in this area of study.

1.3 JUSTIFI

CA

TI

ON OF STUDY

Modeling injection into geothermal reservoirs with the permeability structure of Olkaria

formedthe primary objective of this study. However this could only be done after a

general purpose simulator had been developed. This required allot of mathematical and

computing resources and could only be justified if ultimately the simulator found broader

use than in Olkaria alone. The geology of the recently developed fields in East Africa

indicate that this simulator will find use in these fields which share the same structure as

Olkaria.Furthermore, since layered reservoirs are a more complicated form of

non-layered geothermal systems, this simulator once developed can also be used to simulate

CHAPTER 2 LITE

R

ATU

R

E

R

EVIEW

2.1 GE

N

E

RAL OVER V

I

EW

An indepth and up to date analysis of the state ofgeothermal reservoir modeling has

been presented by O'Sullivan et al. (2005). This paper gives a chronicle of the growth of

geothermal reservoir modeling techniques over theyears and an overview of the

comparative success of the use of numerical simulators in geothermal resource

management.Italso highlights areas of modeling that have received leading attention in

recent years and identifies new trends in geothermal research. Among the leading areas of

geothermal research listed is the understanding of the interaction between the rock

matrix,which forms the bulk of the reservoir, and fractures that run through them. The

interchange of fluids between these two forms of permeability and porosity, and how this

interchange affects the behavior of the reservoir is an area that is being intensely

investigated. This aspect of fluid movement within the reservoir is fundamental to

reservoir description and the uncertainty in the mathematical description of the

interchange and the form of the flow regimes is identified in this paper as aleading area

,.

of geothermal studies.

Amongthe key issues in fracture flow research is the understanding of how fracture

directions and dimensions affect reservoir performance in general and injection profiles

in particular. The mathematical formulation of the problem, also known as the double

porosity, is based on the assumption that the matrix is the primary storage in the reservoir

while fractures actasconduits for transmitting fluid (O'Sullivan et al. 2005; Grant et al.

the interconnections between the two types of porosity, to quantify fluid transfer between

the matrix and fractures. The possibilities in fracture dimensions and orientation and their

effect on fluid flow regimes are so large that it is unlikely that a single model to describe

all ofthem at once can ever be developed. However, given a specific fracture network,

deduced say from field tests, it is possible to develop a description for the system and

reduce the amount of guess work in specific situations. Matrix-fracture interaction

formed thecore of this research.

Thiswork extends the definition of facture from mere fissures in rocks to include contact

points between formations of different geological types. These fractures will have a

larger areal extent than regular fractures and will typically cover most of the field as seen

in Olkaria (Svanbojomson et al. 1983). Because these fractures are of relatively small

size they will not affect the reservoirs stored fluid and energy but will affect fluid

deliveryand particularly return rates of injected brine.

Identifying and recognizing the effect of fractures in geothermal systems occurs at three

stages (Home, 1995; Mathews and Russell, 1967); i) Analysis of pumping and production

tests at well completion, ii) Interference tests, and iii) through injection and tracer tests. I"

The first test yields quantitative values of the fracture and matrix parameters. Since fluid

flowis easier and faster in fractures than in the matrix high production rates generally

indicate the presence of fractures in geothermal systems. Interference tests do provide the

same information as completion tests. However, data from such tests provide quantitative

properties of fractures and matrix, boundaries and discontinuities in reservoir properties

over a wider area and therefore give fields average bulk property (Home, 1995). Injection

enablesisolation of specific fractures. However, it is a lot more difficult to quantify

fractureproperties from injection tests than in the other tests. Doing so requires theuse of

anumerical model. Literature on matrix-fracture interaction can also be classified along

the three parts listed above. In the following review we look at these aspects but pay

special attention to the features of layered reservoirs exemplified by Olkaria geothermal

field.

2.

2 PREVIOUS STUDIES OF OLKARIA GEOLOGY

Studies ofthe hydro-geological structure ofthe entire Olkaria geothermal field based on

secondary minerals, stable isotopes and fluid chemistry by Omenda and Karingithi (2004)

concluded that the fluid movement within the field is controlled by fault systems that

cross the field and lithologic contacts between different geologic formations. In an in

depth study that used data from more than one hundred (100) wells from all parts ofthe

field,the authors concluded that the subsurface geology of Olkaria field consists of a

sequence oftuffs, rhyolite, basalt and trachyte. Surface rocks in Olkaria, 0-500 meters,

consist oftuffs and trachytes. The later are also called upper trachytes to distinguish them

from similar rocks found at deeper levels in the main reservoir. Below the upper trachyte is an extensive layer of basalt of variable thickness (~100 m) covering the entire field.

This layer forms the caprock for the reservoir. Below the caprock, the upper part of the main reservoir is formed by a sequence of rhyolite (700-1000 m), and the lower trachyte

that extends to great depth dominates with intercalations oftuff, rhyolite, and basalt

(Omenda 1994, 1998). The lower trachyte forms the main reservoir rock while the

Thefindings of these studies are consistent with those done using a smaller number of

wells in the early stages of the field's development by Svanbojomson et al. (1983) and

Odongo(1986). Svanbojomson et al. (1983) reviewed the geology of the eastern part of

thefield and concluded that Olkaria consisted of a sequence of near horizontallavas and

tuffs oftrachytic composition. They also recorded a layer of basaltic andesites that is now

known to.bethe fields caprock and referred to simply as basalt by other workers. Ayear

later Odongo (1986) analyzed the geology of the field based on well petrology and

lithology, and surface mappings, and concluded that the layered nature of the field is due

tothe volcanic history of the field which was also consistent with the regional volcanism.

Both these studies identified the contact points between the horizontal lavas as important

sources of permeability.

2.3 NUMERICAL SIMULATION OF OLKARIA

Though the importance of the contact points in fluid delivery in Olkaria reservoir has

been known since the early stages of field development they have not been adequately

incorporated in numerical simulation of the field. A natural state model of Olkaria

geothermal field was recently updated by Ofwona (2003 and 2002) using TOUGH-2

simulator. This model was developed so as to upgrade the pre-exploitation model of the

north east sector of the reservoir by incorporating well data from a number of (then)

recently drilled wells. The simulation also investigated the reservoir performance under

power production at various capacities.

In

productive reservoir covered an area of 120 km", and thickness of 1850 m below a 700

meter caprock. The reservoir was partitioned into 158 blocks (control volumes) in five

vertical layers of which the lower most was the principal reservoir. It also incorporated

themajor hydro-geologic features of the field such as the up flow zones and the

boundaries. In many ways this was an improved model over previous studies done by

Bodvarsson and Pruess (1987) and Bodvarsson (1994).

Thecontrol volumes were assigned hydrologic properties based onwell test and field

matching results adjusted according to results. This arrangement in effect overlooked the

contact points between the formations themselves as points of additional permeability.

Beinglargely a simulation for the natural state with estimates of power potential being

most crucial the simulation considered long term production in a general sense by basing

power estimates on proven field size and power rates per acre. In this regard the simulator

wassimply an improved mass and energy in place. In such models hydraulic properties

arecaptured within an order of magnitude and indeed fundamental parameters like

overall energy in place, the bulk of which is in the rock, are not sensitive to these

properties. Thus, though the model reproduced the static temperature and pressure in the

field,and estimated the recharge rates from the Olkaria fault and the surrounding areas, it

did not give indications of specific trends such as decline rates, enthalpy transients and

I'

phasechanges in the field during exploitation. This is a major omission in the simulation

fora field that was going to be sustained by brine re-injection.

Ofwona (2004a) later undertook a more detailed study of Olkaria I field using a variety of

analytical and computational techniques. These studies were intended to look for ways of

extending the generation life of the existing field while considering the new

developments taking place in the north eastern part of the field. It reviewed the geological

structure of this part of the field based on deepened wells, and injection and tracer tests.

One crucial observation was the impact of the layered nature of the field had on well

productivity based on a shallow well, OW-5, that had been delivering steam to the power

plant for over fifteen years. The mass production from OW-5 increased from 10 tonnes/

hr to 40 tonnes/ hr after deepening from 900m to 2000m (Ofwon a, 2004a). It was also

observed that wells drilled to 2000m and below had higher steam production than wells

whose source of production was in the 1OOOmto 1200m interval.

In view of these results Ofwona (2004b) concluded once again that the layered nature of

the reservoir was responsible for the increase in steam production of the well that had

been deepened and re-confirmed the importance of the contact points between the layers

as responsible for the increased steam production. The study therefore recommended new

completion depths for the production wells so as to produce from the deeper zones. In

spite of this significant confirmation of the importance of the contact points as sources of

additional production, Ofwona (2004b) proceeded to analyze the power potential based

only on the mass in place and averaged well delivery. Thus the immediate experience of

the ability of the contact zones to deliver fluids to the wells was not incorporated in the

analysis.

In a related study, Ouma (2002) presented an investigation of brine re-injection strategy

in the North-East sector of Olkaria. By considering results from five injection and tracer

tests, and one interference test between 1996 and 2004, Ouma analyzed the suitability of

various re-injection wells and injection options for the field. The analysis of results was

analytical and used averaged values of hydrologic parameters. It considered injection

issues generally without taking into consideration the in-homogeneities in the field either

used by Ofwona (2002) was not addressed nor the lithologic and stratigraphic controls

raised by Omenda and Karingithi (2004) included. It is the view of this'study that this

constituted a major omission in the analysis.

An important study of injection and tracer tests in Olkaria geothermal field was reported

byMwawongo (2004). This study analyzed results of injection tests done between July

1995 and September 1997 in the older part of the field. Flourescein sodium was the tracer

in this study. This analysis represents the first attempt at estimating hydrologic properties

and dimensions ofthe formations from injection tests. This part of the field is traversed

byawell defined regional fracture that runs in the north-south across field. Estimates of

hydraulic properties were crucial as injection studies were meant to revive wells in the

central part of Olkaria field some which had experienced complete dry-out. This part of the field had experienced the highest pressure draw down resulting in several wells being

retired due to loss of production (Kariuki 2004, Mburu 2003). Substantial benefit had

been realized by brine injection on the perimeter of the field which had been ongoing for

adecade at this time. It was therefore expected that direct injection into the field center

which h_,.. ad experienced severe drawdown would even lead to better results.

In his analysis Mwawongo (2004) used both analytical and numerical models to match

the field data. A tracer inversion program TRINV developed by Arason et al. (2004) was used to do the analytical part while the numerical analysis used the TOUGH-2 simulator

that had been used to match other tests previously. In the analytical model only single

parameter hydrologic values were used to estimate the tracer return times and the

cumulative values of tracer over the entire tests. This assumed that the tracer was

constant throughout. This assumption was in contradiction to observations in the field test

itself where practically all the profiles were multi peaked indicating several return

channels. Thus these results at best represent average hydrologic parameters of the

assumed paths.

Though this model used improved and better method than the previous ones it still did

not adequately a?dress all the issues. For example it.used a one dimensional flow channel

between the injection well and each of the production wells even when one dimensional

models can only be considered approximations of reservoirs where fluid flow is subject 0

gravity and therefore 3-dimensional. This one dimensional model was a channel

representing a fracture embedded in regular matrix. Since only one well received

substantial amounts of tracer the simulation ignored most of the field results. For this

model to match the tracer profile it was necessary to use a fracture with the following

dimensions; 800 meters deep, 400 meters wide and 2 meters thick and a porosity of 50%.

Though the matches were spot-on with field data the values used to obtain them are

unrealistic and raises suspicions about the correctness of the methods used. For example

it is improbable that a fracture that is 2 meters thick, 400 meters wide and 800 meters

deep would be 50% porous. Such a formation would not be physically stable under the

over burden alone. Olkaria geothermal system is composed of high density volcanic

rocks whose measured porosity average 7% and do not exceed 15% even for the air filled

pumice. Fracture porosities are even less than matrix porosity and enthalpy transient

studies show that values can not exceed 2% (Bodvarsson and Pruess, 1987; Grant 1979).

Further more if such an extensive high porosity fracture system existed in Olkaria the

the wells produce an average of2.5 Mwe of steam, well below the world average of 4 to

5Mwe per well. Another weakness of the analysis by Mwawongo was not to consider the

re-saturating effect of injection in a fully flushed steam zone.

In arelated study Manyonge (1997) also developed a numerical model for Olkaria

geothermal field using TOUGH simulator, the predecessor of TOUGH-2. This model

investigated the capacity of Olkaria field to sustain combined power generation from the

existing 45 Mwepower plant and the then upcoming 64 Mwe power plant in Olkaria

northeast part ofthe field. The simulation used a coarse 24km X 24 km grid divided into

blocks of 2km by 2km, and vertical profile of 8layers of variable thickness over a depth

of 1800m. Because ofthe large size of the simulated area, 576 km'', compared to proven

area== 1Okm",boundary conditions were only partly captured by this simulation. The

simulation reported pressure and enthalpy changes over atwenty five (25) year period. It

did not consider brine re-injection.

The division of the reservoir into eight (8) layers in the vertical direction represented a

significant refinement over all the previous models. However, this refinement did not

result jn any improvement since these layers, and in fact all blocks in the field, were

assigned exactly the same (constant) porosity and physical dimensions. This is contrary

tofield test experience where tracer profiles were multi-peaked and significant variation

in production rates and discharge enthalpy between wells in the field indicated

in-homogeneity. The many layers in the simulation only introduced time lags in the

propagation of pressure without altering their magnitude in any substantial manner

(Azziz and Settari, 1979). The coarse grid also represented another weakness in the

model.With an areal extent of 4 km2per block, an area equal to the old part of the field,

allthe twenty three (23) wells in this part of the field were represented by one "big well"

with one flow rate and enthalpy. Thus, effectively the simulation was reduced into a

lumped parameter model instead of the distributed parameter model that was intended.

Thissituation was not improved by the fact that the model assumed a perfectly porous

media with a permeability of three (3) Darcys. Assuming a porous media was a draw

back since all models had used double porosity model of which Olkaria was

representative. Furthermore completion tests in all Olkaria wells gave permeabilties that

were in the range of hundreds ofmilli Darcy, an order of magnitude less than what the

model assumed. This fact is supported by the modest average production rates of wells,

2.5 Mwe.

Because of the large grid size, perfect porous media assumption, representation ofthe

entire Olkaria east field byone well and the high permeability, the preceding study did

notyield realistic results. For example the high permeability led to only minor enthalpy

transients being predicted, 1350 kjlkg. This is in contradiction to field experience where

observed enthalpies are above 1500 kjlkg. The simulation also predicted only minor

pressure drops for most ofthe eight (8) layers, 0.3 bars and a maximum of 15 bars for the

central layers. These values are small because high permeability implies high recharge

and therefore better pressure sustenance while the exploited area was only 3% of the

entirefield. Finally by not considering brine injection the simulator can provide no

meaningful in sight to the field management options now that Olkaria injects 100% ofthe

brineand blow-down which all total more than 1000 metric tones per hour.

None of the field studies highlighted above considered the additional permeability at the

numerical and analytical studies by Ofwona (2002, 2003, 2004a, 2004b) and Mwawongo

(2004), considered this form of hydrologic transport even when well productivity and

tracer profiles did show the effect of multiple flow paths. The outcome of these omissions

is that the simulations have ignored a significant component of permeability which has an

important impact on fluid return characteristics of injected fluids.

To some extent these omissions can be attributed to the tools available to the

investigators. TOUGH-2 simulator is designed formodeling multi-phase flow of fluid

and heat in porous media. It implements fluid flow in fractures by using the double

porosity model as developed by Warren (1963) modified by the Multiple Interacting

Continua, MINC, Narasirrthan(1982). In this formulation the matrix acts as the storage

for fluid while the fractures act as the media for transport. Fluid movement only occurs

from the matrix to the fracture and not vice versa. In practical implementation the

interaction between fractures and matrix is'implemented by use of geometrical factors

that defines the areal extent of common area between the fractures and matrix.

This limitation of the flow direction from matrix to fracture generally meant none of the

injected fluid could travel in to the matrix from the fractures. The aim of this study was to

I'

investigate the effect of multi path in geothermal reservoirs. Numerical modeling is in

effect an extensive mathematical formulation of the problem of quantifying the response

and behavior of geothermal systems. Geothermal systems are complicated systems where

a large number of variables form the input parameters that can affect its characteristics at

anyone time. The first step in numerical modeling of injection into geothermal systems is

therefore the mathematical formulation of equations that apply for the flow of fluid and

heat in geothennal systems. Modeling injection is therefore only a special case of general

geothennal modeling and must draw from the basic tools of geothermal modeling.

In the early 1980's workers in Lawrence Beckley Laboratory took the first steps towards

incorporating effects of fractures in analysis of injection tests. In their work they

considered reservoirs with large vertical fractures and derived the relevant equations

governing heat transfer (Bodvarsson et al. 1986). The analysis did show a lower thermal

. .

efficiency caused by fractures. They were able to incorporate fractures into their

numerical package MULKOM which was then still under development.

This research was undertaken with the specific objective of modeling reservoirs with

significant horizontal permeability in relatively thin zones as a refinement and

improvement to the existing interpretive tool in studies of re-injection. This meant that

governing equations had to be derived and later translated into a computer program to

CHAPTER

3 T

HE

OR

Y

3.1 MOD

ELlNG OF GEOTHERMAL SYSTEMS

From the foregoing analysis it follows that modeling brine injection into geothermal

reservoirs is a specialized form of general numerical modeling. Modeling injection in

layered geothermal systems begins by first developing a general purpose numerical

simulator for modeling conventional geothermal systems after which it should

incorporate the additional processes that take place in reservoirs with the permeability

structure of a layered system. This calls for the appropriate modification of the mass and

energy balance equations to account for fluid transfer between different formations. The

model that was developed here had a special property of being applicable to layered

reservoirs and the equations had to be developed to cover this scenario. The fundamental

difference in layered reservoirs is in the geological construction in which the additional

permeability occurs at the contact point between layers rather than within the layers only

as is the case in many reservoirs. The theoretical distinction between this model and

double porosity models is the fact that this model allows for fluid flow in both directions

in the matrix-fracture network while double porosity models fluid flow can only occur

from the matrix to the fracture and not the other way round (O'Sullivan et al., 2005).

Modeling geothermal systems as stated earlier is a practice that draws from the

mathematical description of a system or process so as to represent and predict the

behavior of the actual physical system. However the process must first begin with the

understanding of the properties that control fluid flow in the system. Thus the first step in

the field. Geological models are based on both physical geology as seen on the surface,

correlations of lithologies from well cuttings, and the known regional and local features.

These models must identify the physical features that control fluid and energy transfer in

the reservoir and their variation with location. The models should provide the frame work

on which all other issues in the field are based.

One short coming of geological data is the non-availability of data at all points within the

reservoir. This makes geologic information perpetually incomplete. The sub-surface

reservoir and the rocks contained in them are only visible through drilled wells which

cover only a limited part of the reservoir. Production wells are normally drilled in

selected parts of the field based on the power production potential and have to be

adequately spaced to avoid interference during production. Further more wells penetrate

only a limited depth of reservoirs leaving a greater part of the reservoir unknown. This

puts a limit at how much data can be collected from a field. In addition there is little or no

interest in drilling the cooler non-productive geologic formations surrounding reservoirs

with which the reservoir is continuously interacting. This leads to gaps in input data

particularly the boundary conditions. The outcome is that geological data is available

only for areas that are small compared to the total reservoir volume. This makes

lithological correlation difficult and is rarely achieved. To overcome this shortage a

substantial part of the geological model has to be inferred or interpreted which introduces

uncertainties and inaccuracies in the model. This study relied heavily on data from

Key

o

Recharge

Outflow

.•...

Quaternary lavas Basalt

Trachyte/Rhyolite Mau Volcanics (Tuff) Intrusive

c

::

::

:

'

>

•

EOF East Olkaria Field

NEF North East Field OWF Olkaria West Field

Mau

Escarpment

2

3

4

I I 2km

Figure 3.1 Geology ofOlkaria geothermal field (Omenda, 1994)

3.2 OVER VIEW OF THE PROBLEM

Equations for two phase fluid flow in geothermal systems have been presented in several

references (Barenblatt et al. 1960; Pruess, 2002, 1992 and 1985; Aziz and Settari 1979). The number of equations that are applicable to a geothermal system atanyone time

depends on the number of mobile fluid phases in the reservoir and whether energy

balance considerations have to be incorporated or not. These equations are generally

complicated. The equations represent diffusive processes and are of the parabolic type.

Because of phase change and dependence of fluid flow properties on temperature and

pressure these equations are coupled and non-linear. Geothermal systems are fractured

and in-homogenous which complicates the problem further.

The governing equations for a two phase geothermal system are three; two mass balance

equations for steam and water respectively, and an energy balance equation. Solutions to

these equations will specify the evolution of the thermodynamic parameters that govern

the movement of fluid and heat in the reservoir. An additional equation will be required

for injection tests with tracers. The complexity of the tracer equation will depend on

whether the tracer partitions in all phases within the reservoir, adsorbs and desorbs from

the rock surface, reacts or decays with temperature and time (Adams et al., 1990.Pruess,

2002). Commonly used tracers are only soluble in the liquid phase and will therefore

need only one additional equation. That was the case for the Flourescein sodium and

potassium iodide tracers in this study.

In-homogeneity of geothermal systems implies that the coupled governing equations can

only be solved in domains in which some homogeneity can be assumed. These domains

are also called control volumes or blocks. Thus after constructing a geological model the

reservoir must be partitioned into separate volumes in which uniform properties can be

assumed. Since the volumes are in contact with each other the equations have to be

solved simultaneously to give the thermodynamic conditions of the reservoir at the same

time. It is impossible to obtain solutions to these equations in closed form for this kind of

Three leading methods of numerical solutions are Direct Finite Difference, Integral Finite

Difference and Finite Elements. These methods are nearly equivalent, however each of

these three methods is best suited to specific problems. These methods reduce partial

differential equations to discrete difference equations that are defined only in certain

domains. Bydoing this, the problem of solving for the unknowns is transformed into an

algebraic problem.

Direct finite difference is the simplest ofthese methods. The derivatives are replaced by

their equivalent discrete forms. This method can only be used to solve partial differential

equations in Cartesian coordinates. It is therefore suitable for problems that have simple

geometric forms, a situation rarely seen in geothermal systems. This limits the use of

solutions developed this way.

For integral finite difference equations are discretized by first taking the volume integrals

of the partial differential equations. In this method reservoir properties are averaged over

the control volume. One advantage of this method is that it is possible to convert a

number of derivatives in the equation from volume to surface integrals using Gauss

diver-gence theorem. This reduces the order of the partial differential equation by one,

which, in turn reduces the truncation errors in the eventual calculations. This method can

be applied to problems in any coordinate system including situations where control

volumes are covered by irregular surfaces.

Finite elements are fairly recent adaptations to fluid flow problems from civil engineering

where they are routinely applied to problems of structural analysis. Finite elements like

coordinate system. In addition the surfaces outlining the volume can take irregular forms.

Fluid flow into and out of the volume does not have to be perpendicular to the surface as

is the case for the other two methods. This method is therefore best suited to problems

where there are uncertainties in representative values of the geologic properties. However

the method uses averages of known values on the perimeters of the volume to estimate

the values of the properties within the volume. This is done through the use of weighted

functions that may introduce errors of their own in the computations. This is a drawback

the other two methods do not have and is not required for the problem in question.

In this study the integral finite difference method was ~elected. This is because the

problem as defined, the fluid flow parameters were known and the use of finite elements

was unnecessary. This method was also going to give results that will be applied to any

volume configurations irrespective of the surface area and orientation. The direct finite

difference was unsuitable for this problem as it limits geometric systems to which the

solution can be applied.

In the following section we shall present the applicable equations to numerical solution.

The ~quations shall also be discretized in preparation for developing a solution of the

problem.

3

.

3 FUNDAMENTAL EQUATIONS

The integral form of the mass balance equation obtained by equating surface flux across

all surfaces over a control volume to the rate of mass accumulation and sources for each

phase is given by,

(3.1)

In equation (3.1)iis the phase i.e. water or steam, p is phase density, f.1i isviscosity, P

.' .

ispressure, Siisphase fraction ( or saturation), his height above reference point, gis

gravity constant, d A and dV are surface and volume elements respectively, while

k

k

where liquid and gas phases are at equal pressure is assumed. The flux terms are obtained

directly from Darcy's equation.

The energy balance equation for the control volume is given bythe equating the sum of

energy fluxes for all mobile phases across the surface to the rate of energy accumulation

within the volume plus sources. The single equation representing this is given below,

The first term under the integral on the left hand side of equation (3.2) is the sum of energy flux inall mobile phases in the reservoir. It is obtained directly from the product

ofthe mass flux given by 3.1 and the enthalpy for the phase,

hi

'

phase fraction, fjJ is porosity, k.]: is product of relative permeability and permeability,

KT is thermal conductivity, URis internal energy for the rock,

Q

i

point. The subscript refers to the phase under consideration. The rest of the terms in this equation have the same meanings as in equation (3.1).

The integral form of the equation for tracer flow is given below,

(3.3)

Cistracer concentration, D is diffusion coefficient (diffusivity), the sum of both

molecular and fracture dispersion,

q

Q

c

is the sum of all sources in the control volume. Equation (3.3) depends on fluid flux,

obtained by the application of Darcy' s equation to the pressure equation, (3.1).

Except for the energy flux terms in equation (3.2), equations (3.1) to (3.3) have the same

basic form. This was exploited in the development of the numerical solution. The

fundamental equations, with minimal change of terms, can therefore use the same solver. For this reason the equation solver was developed for the pressure equation first, then

later modified to solve the energy and tracer flow equations. Inherent in equations (3.1)

and (3.2) is the saturation which gives the volume fraction of two phases. Saturation was

calculated directly by re-arranging one of the mass equations.

As stated earlier geothermal systems are not homogeneous so the equations have to be

solved for regions over which homogeneity in formation properties can be assumed. In

these control or finite volumes as they are called, the equations use average values of the

fluid and formation properties. By doing this the equations are effectively a discretized.

3.4

DISCRETIZATION OF EQUATIONS

This section reviews the method of solution used in solving the discrete equations. The

presentation uses the pressure equation to show how this was achieved and later exploits

thesimilarity of the pressure equation with other equations to present without detail the

discretized forms of the other two equations.

The left hand side of the mass equation, (3.1), is a surface integral which is obtained by

summation of surface fluxes around each block. The expressions for pressure gradients

were approximated bythe quotient ofthe pressure difference between the block under

consideration and the neighboring block to the inter block distance between the blocks.

Introducing the new expressions for surface integrals and derivatives the left hand side of

the mass balance, the equation, (3.1) becomes,

In equation (3.4)Pj is the average pressure of the block under consideration.F; is the ,.

average pressure ofthe next block, l'ukj is the inter block distance between blocks

j

k ,

performed for all k, which is the sum of fluxes from all blocks next to

j .The

term on the right hand side of equation (3.4) represents effects of gravity, as a result of

the gradient in the vertical direction or height, h. In Cartesian coordinates the gradients on

the x-y plane are zero while ones along the vertical axis are opposite each other and

The time derivative of the mass accumulation term on the right hand side of equation

(3.1) can be written as follows,

a

f

a

- rp(p,S,)dv

=

at

at

This isthe change in mass within the control volume V. This can be calculated directly

from mass difference due to change in saturation, p~essure or density at two different

times. This is written as,

(3.6)

The density term depends directly on pressure and can be estimated at each time step.

This dependence can be used in the calculation of mass change using the equation for

slightly compressible fluids,

ap

i

op,

er

er

-=--=Cp

.-at

ap at

at

(3.7)

where c is compressibility for the fluid under consideration. Equation (3.7) can be

simplified further to,

ap

~

P(r,t

+

/

:)

.

t)- Per, t) ~

r

:

'

_

P

"

cp -

=

cp

.

=

cp

.

----,

a

t

'

/

:)

.

t

'/

:)

.

t

where the superscripts represent two successive time steps. The two forms of mass

which one to use depends on type of information available. Steam table values are best

suited for (3.6) but mathematical correlations work best for (3.8). Equation (3.8) was

adopted for this problem. Leaver correlations (see Appendix D) that give properties of

steam and water in terms of temperature and pressure were used in this project. Thus the discrete form of equation (3.1) can be written as,

ok

:

k(R

-P)

k k

r'"

_pn

~""'i rt kA•• 1.dL1'i+anI2~dL1'i

d.SV

Q

L.

.

:

.

f.t

+

k ~ ~ ~ ~~

(3.9)

Sources and sinks are given by the term

Q

into or out of the region or control volume per unit time, in the same units as St .

Common examples of sources and sinks in geothermal systems are production and

injection wells. The first step in solving equation (3.9) is the selection of the time steps at

which the terms on the left hand side of the equation are to be evaluated. The fully

implicit form where all the terms are evaluated at the next time is adopted in this solution.

Then equation (3.9) takes the form below.

p n+!

=P

.

+Q

up,low

(3.10)

n

k t- ~ P

.

u

:

«:

m

11 uplow

All terms on the right hand side represent the values at the current time step and are

therefore known. Equations similar to equation(3.11) is given for each control volume in

the reservoir. These equations define a complete statement of the problem. These

equations are coupled since several pressure terms and other parameters appear in several

equations at the same time. They must therefore be solved simultaneously.

Equation (3.11) isthe governing equation for pressure in each control volume or block. It

is acomplicated equation since it includes values of pressure from many blocks. It is

heavily non-linear since all the coefficients of values of pressure are made up of products

of density, viscosity and compressibility. All these parameters depend on pressure and

have to be calculated at the same time with the actual pressure. Using similar arguments

the energy balance equation can also discretized as follows. The right hand side of

equation (3.2) can be written in the following form,

(3.12)

up,lOWj